L11a404

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_10,13,5,14 X_8,17,9,18 X_14,7,15,8 X_18,9,19,10 X_22,20,11,19 X_20,16,21,15 X_16,22,17,21 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -3}, {11, -2, 3, -5, 8, -9, 4, -6, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 - v w u 4 + 3 v 2 u 3 -5 v u 3 - v 2 w u 3 + 4 v w u 3 -2 w u 3 + 2 u 3 -4 v 2 u 2 + 7 v u 2 + 2 v 2 w u 2 -7 v w u 2 + 4 w u 2 -2 u 2 + 2 v 2 u -4 v u -2 v 2 w u + 5 v w u -3 w u + u + v - v w + w$ (db)
Jones polynomial	$- q 2 + 4 q -9 + 14 q -1 -19 q -2 + 22 q -3 -20 q -4 + 19 q -5 -12 q -6 + 8 q -7 -3 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + 2 a 8 z -2 + 2 a 8 -3 z 4 a 6 -9 z 2 a 6 -5 a 6 z -2 -12 a 6 + 2 z 6 a 4 + 8 z 4 a 4 + 16 z 2 a 4 + 4 a 4 z -2 + 14 a 4 + z 6 a 2 + z 4 a 2 -2 z 2 a 2 - a 2 z -2 -4 a 2 - z 4 - z 2$ (db)
Kauffman polynomial	$z 6 a 10 -3 z 4 a 10 + 3 z 2 a 10 - a 10 + 3 z 7 a 9 -7 z 5 a 9 + 4 z 3 a 9 + 5 z 8 a 8 -11 z 6 a 8 + 10 z 4 a 8 -11 z 2 a 8 -2 a 8 z -2 + 8 a 8 + 4 z 9 a 7 -18 z 5 a 7 + 25 z 3 a 7 -18 z a 7 + 5 a 7 z -1 + z 10 a 6 + 16 z 8 a 6 -50 z 6 a 6 + 62 z 4 a 6 -45 z 2 a 6 -5 a 6 z -2 + 20 a 6 + 9 z 9 a 5 -3 z 7 a 5 -34 z 5 a 5 + 53 z 3 a 5 -33 z a 5 + 9 a 5 z -1 + z 10 a 4 + 20 z 8 a 4 -55 z 6 a 4 + 62 z 4 a 4 -38 z 2 a 4 -4 a 4 z -2 + 15 a 4 + 5 z 9 a 3 + 8 z 7 a 3 -37 z 5 a 3 + 41 z 3 a 3 -19 z a 3 + 5 a 3 z -1 + 9 z 8 a 2 -13 z 6 a 2 + 8 z 4 a 2 -6 z 2 a 2 - a 2 z -2 + 3 a 2 + 8 z 7 a -13 z 5 a + 8 z 3 a -4 z a + a z -1 + 4 z 6 -5 z 4 + z 2 + z 5 a -1 - z 3 a -1$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-2 is the signature of L11a404. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$